When the function \(F\)  is represented as an expectation \(F\left( x \right) = {E_\xi }\left[ {f\left( {\xi ,x} \right)} \right]\) , as in our problem, we need to use a stochastic version of FISTA [1]}, [2]}. The true gradient \({\nabla F({y_k})}\)  is replaced by a sample-path gradient \(\nabla f({y_k})\)  in each iteration. When used with a constant step size, the stochastic FISTA achieves a convergence rate of \({\rm O}\left( {1/\sqrt{k} } \right)\), and with decreasing step sizes it achieves a rate of \({\rm O}\left( {\log \left( k \right)/\sqrt{k} } \right)\). Moreover, the stochastic FISTA is close to its \({\rm O}\left( {1/{k^2}} \right)\)  deterministic behavior in the first several iterations [2]}.
